Perturbing Masses: A Study of Centered Co-Circular Configurations in Power-Law n-Body Problems

This research investigates centered co-circular central configurations in the general power-law potential $n$-body problem. Firstly, there are no such configurations when all masses are equal, except for two; secondly, unless all masses are equal, no such configurations exist when masses can be divided into two sets of equal masses. We adapt Wang’s criterion and incorporate insights on cyclic quadrilaterals, alongside mathematical induction.